Holomorphic flat projective structures on projective threefolds
type: Item
instance of: scholarly article
title: Holomorphic flat projective structures on projective threefolds
zbMATH Open document ID: 0831.32014
DOI: 10.1007/BF00773656
author: E. Ballico
published in: Annals of Global Analysis and Geometry
publication date: 1995-10-09T00:00:00Z
review text: The author intends to show the usefulness of the recent methods of algebraic geometry (Mori theory) to the study of differential geometric structures on algebraic manifolds. Main Theorem: Let \(X\) be a smooth projective threefold with a holomorphic flat projective structure (i.e. a normal integrable projective connection) and Kodaira dimension \(\kappa \neq 1\), 2. Then: (a) \(\kappa<0 \Leftrightarrow X \cong \mathbb{P}^3\); (b) \(\kappa = 0 \Leftrightarrow\) there is a finite unramified covering of \(X\) by a torus; (c) \(\kappa = 3 \Leftrightarrow\) the universal covering of \(X\) is the unit ball of \(\mathbb{C}^3\). Remark: Let \(X\) be (a) a four-dimensional projective manifold with \(\kappa \geq 0\) and \(K_X\) not nef, or (b) a compact complex 3-fold in the class \(C\) with \(\kappa < 0\) and a nontrivial Albanese torus. Then \(X\) has no holomorphic flat projective structure. The main use of Mori theory is to show that suitable projective varieties contain a rational curve and to deform it inside the ambient projective variety.
Mathematics Subject Classification ID: 32J17
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 53B10
Mathematics Subject Classification ID: 53C99
zbMATH DE Number: 792875
zbMATH Keywords: smooth projective threefold
zbMATH Keywords: holomorphic flat projective structure
zbMATH Keywords: Mori theory
reviewed by: Krzysztof Dabrowski
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00773656
OpenAlex ID: W2059856872
description: scientific article; zbMATH DE number 792875